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Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\\mid b_0$, but $p^t\\nmid b_{i}\\ {\\rm for}\\ 1\\leq i\\leq m$.\n  Then, we prove that there is no infinite subset $\\mathcal A$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+...+a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\\ a_{i,j}\\in \\mathcal A$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Ru\\'{e} for bilinear case, and answers a question pose"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.1920","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-09T13:14:53Z","cross_cats_sorted":["math.CO","math.CV"],"title_canon_sha256":"119a73b047343c42f9c90bf6169994daba54ee7d5a3256c40e3882fe4201d649","abstract_canon_sha256":"e6cf8991dab3d672c877c2eac3efee9910218a6d0d28688261013f515c07562a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:28.757971Z","signature_b64":"1A7TVbQvZaKphNg9dJMyNm2AqSxPchVM8yZugYerWnRdfOr264FI9mFgetT2ffzsfXlOyFKMS2SmJ1Vw0HjMAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85563055b6ad5a9f252e6c6047ece1907d8723e70d1059eafc78d98f89964aa5","last_reissued_at":"2026-05-18T04:14:28.757331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:28.757331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A question of S\\'{a}rkozy and S\\'{o}s on representation functions] {A question of S\\'{a}rkozy and S\\'{o}s on representation functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.CV"],"primary_cat":"math.NT","authors_text":"Lianrong Ma, Yan Li","submitted_at":"2011-08-09T13:14:53Z","abstract_excerpt":"For $m\\geq 1$, let $0<b_0<b_1<...<b_m$ and $\\ e_0,e_1,...,e_m>0$ be fixed positive integers. 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